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1 Biodiversity Data Analysis: Testing Statistical Hypotheses By Joanna Weremijewicz, Simeon Yurek, Steven Green, Ph. D. and Dana Krempels, Ph. D. In biological science, investigators often collect biological observations that can be tabulated as numerical facts, also known as data (singular = datum). Biological research can yield several different types of data. Important measurements include counts (frequency) and those that describe characteristics (length, mass, etc.). Data from a sample are often used to calculate estimates of the average values of the population of interest (mean, mode, and median) and others describing the dispersion around those values (range, variance, and standard deviation). I. Data, Parameters, and Statistics: A Review Recall that data can be of three basic types: 1. Attribute data. These are descriptive, "either-or" measurements, and usually describe the presence or absence of a particular attribute. The presence or absence of a genetic trait ("freckles" or "no freckles") or the type of genetic trait (type A, B, AB or o blood) are examples. Because such data have no specific sequence, they are considered unordered. 2. Discrete numerical data. These correspond to biological observations counted as integers (whole numbers). The number of leaves on each member of a group of plants, the number of breaths per minute in a group of newborns or the number of beetles per square meter of forest floor are all examples of discrete numerical data. These data are ordered, but do not describe physical attributes of the things being counted. 3. Continuous numerical data. These are data that fall along a numerical continuum. The limit of resolution of such data is the accuracy of the methods and instruments used to collect them. Examples are tail length, brain volume, percent body fat...anything that varies on a continuous scale. Rates (such as decomposition of hydrogen peroxide per minute or uptake of oxygen during respiration over the course of an hour) are also numerical continuous data. (Figure 1). (Continuous numerical data generally fall along a normal (Gaussian) distribution. This distribution is a function indicating the probability that a data point will fall between any two real numbers.) When an investigator collects numerical data from a group of subjects, s/he must determine how and with what frequency the data vary. For example, if one wished to study the distribution of shoe size in the human population, one might measure the shoe size of a sample of the human population (say, 50 individuals) and graph the numbers with "shoe size" on the x-axis and "number of individuals" on the y-axis. The resulting figure shows the frequency distribution of the data, a representation of how often a particular data point occurs at a given measurement. Biodiversity Data Analysis 1

2 Usually, data measurements are distributed over a range of values. Measures of the tendency of measurements to occur near the center of the range include the population mean (the average measurement), the median (the measurement located at the exact center of the range) and the mode (the most common measurement in the range). It is also important to understand how much variation a group of subjects exhibits around the mean. For example, if the average human shoe size is "9," we must determine whether shoe size forms a very wide distribution (with a relatively small number of individuals wearing all sizes from 1-15) or one which hovers near the mean (with a relatively large number of individuals wearing sizes 7 through 10, and many fewer wearing sizes 1-6 and 11-15). Measurements of dispersion around the mean include the range, variance and standard deviation. Parameters and Statistics If you were able to measure the height of every adult male Homo sapiens who ever existed, and then calculate a mean, median, mode, range, variance and standard deviation from your measurements, those values would be known as parameters. They represent the actual values as calculated from measuring every member of a population of interest. Obviously, it is very difficult to obtain data from every member of a population of interest, and impossible of that population is theoretically infinite in size. However, one can estimate parameters by randomly sampling members of the population. Such an estimate, calculated from measurements of a subset of the entire population, is known as a statistic. In general, parameters are written as Greek symbols equivalent to the Roman symbols used to represent statistics. For example, the standard deviation for a subset of an entire population is written as "s", whereas the true population parameter is written as σ. II. From Raw Data to Index of Biodiversity Now that you ve had a chance to review a bit of statistical information, it s time to apply it to your own project. In this section, you will be guided through the process of calculating indices from your raw data collected over the past two weeks, and then using those indices to compare the two habitat types you chose. A. Ordinal Data Points: Menhinick s Index (D) When you collected and counted organisms in your samples, you were taking a survey of the number of different species present in each of your two habitat types. You counted the number of individuals of various species in 12 samples collected from each of your two selected habitat types. From these counts, you can calculate a Menhinick s Index (D) for each counted sample. At the end of your preliminary calculations, you should have ten D values for each of the two habitats you are comparing. You will use these D values in the Mann-Whitney U test to determine whether your two habitats differ significantly in the measure of biodiversity you have chosen (species richness). Recall the formula for Menhinick s index, which represents the number of species in the sample divided by the square root of the number of individuals in the sample. s = the number of different species in your sample N = the total number of individual organisms in the sample. Biodiversity Data Analysis 2

3 Your team should have counted at least 10 samples from each of your two habitats, and can now calculate one Menhinick s index (D value) for each sample. Tabulate your D values here: Sample # D habitat1 D habitat So what do we do with these indices? You may have an intuitive sense that they will allow you to determine whether your two sampled habitats overlap in their degrees of biodiversity. But science isn t about intuition. Statistics and statistical tests are used to test whether the results of an experiment are significantly different from the null hypothesis prediction. What is meant by "significant?" For that matter, what is meant by "expected" results? To answer these questions, we must consider the matter of probability. B. Probability The significance level (also known as alpha (α)) for a given study is set by the investigator before the analysis is begun. Alpha is defined as the probability of mistakenly rejecting a null hypothesis that is true (Type I error). By convention, α is usually set at 0.05 (5%). The probability that an observed result is due to some factor other than chance is known as P. The result of a statistical test is a statistic. For example, the student s t test yields a t statistic, the Chi-square test yields a X2 statistic, and the Mann-Whitney U test yields a U statistic. Every value of a particular statistic is associated with a particular P value. If the P value associated with a calculated statistic (e.g., the U statistic you will calculate with the Mann- Whitney test, to be described below) is 0.05, this means that there is only a 5% chance that the rejection of the null hypothesis will be incorrect. A P value of less than 0.05 means that there is an even lower chance of a Type 1 error. (For example, a P value of 0.01 means that there is only a 1% chance that the results are due to chance, and not to the factor you are examining.) In essence, α is a cut off value that defines the area(s) in a probability distribution where a particular value is unlikely to fall. In some studies, a more rigorous α of 0.01 (1%) is required to reject the null hypothesis, and in some others, a more lenient α of 0.1 (10%) is allowed for rejection of the null hypothesis. For our study of biodiversity, you will use an α level of The term "significant" as used in every day conversation is not the same as the statistical meaning of the word. In scientific endeavors, significance has a highly specific and important definition. Every time you read the word "significant" in this lab manual, know that we refer to the following scientifically accepted standard: Biodiversity Data Analysis 3

4 The difference between an observed and expected result is said to be statistically significant if and only if: Under the assumption that there is no true difference, the probability that the observed difference would be at least as large as that actually seen is less than or equal to α (5%; 0.05). Conversely, under the assumption that there is no true difference, the probability that the observed difference would be smaller than that actually seen is greater than 95% (0.95). Once an investigator has calculated a statistic from collected data, s/he must be able to draw conclusions from it. How does one determine whether deviations from the expected (null hypothesis) are significant? There is a specific probability value linked to every possible value of any statistic. A probability distribution assigns a relative probability of any possible outcome (e.g., Menhinick s Index). The species richness calculations you performed for each sample, while expressed as a number, are not distributed along a normal curve. They are ordinal, rather than continuous, data. For this reason, a non-parametric statistical test, the Mann-Whitney U test, will be employed for your analysis. C. Statistical Hypotheses A non-parametric test is used to test the significance of qualitative or attribute data such as those you have been collecting for this research project. In the following sections, you will learn how to apply a statistical test to your data. Your team should already have devised two statistical hypotheses stated in terms of opposing statements, the null hypothesis (H o ) and the alternative hypothesis (H a ). The null hypothesis states that there is no significant difference between two populations being compared. The alternative hypothesis may be either directional (one-tailed), stating the precise way in which the two populations will differ ( Pond A will have greater species richness than Pond B. ), or nondirectional (two-tailed), not specifying the way in which two populations will differ ( Pond A and Pond B will differ in species richness ). Your team should already have devised null and alternative hypotheses for your survey of biodiversity. To determine whether or not there is a difference in biodiversity between your two sample sites, you must now perform statistical tests on your data, the series of Menhinick s Indices (D) that you calculated from your individual survey samples. III. Applying a Statistical Test to Your Menhinick s Indices Once your team has calculated a Menhinick s index (D) for each of your 12 samples from each of the two habitats, you are ready to employ a statistical test to determine whether there is overlap between the range of calculated indices. If there is a great deal of overlap, it means that there is not a significant difference between them, and you will fail to reject your null hypothesis. However, if there is very little overlap (5% or less), you can confidently conclude that two habitats do differ significantly in their species richness, and reject your null hypothesis. A. Non-parametric test for two samples: Mann-Whitney U The Mann-Whitney test allows the investigator (you) to compare your two habitat types without assuming that your D values are normally distributed. The Mann-Whitney U does have its rules. For this test to be appropriate: Biodiversity Data Analysis 4

5 You must be comparing two random, independent samples (your two sites) The measurements (Menhinick s Indices, in our case) should be ordinal No two measurements should have exactly the same value (though we can deal with ties in a way that will be explained shortly). The Mann-Whitney U test allows the investigator to determine whether there is a significant difference between two sets of ordered/ranked data, such as those your team has collected in its biodiversity study. Here is a stepwise explanation and example of how to apply this test to your data. 1. State your null and alternative hypotheses. (You already have done this, right?) H o : H a : Example: H o : There is no difference in the ranks of species richness between a silted pond and a clear pond. H o : There is a difference in the ranks of species richness between a silted pond and a clear pond. 2. State the significance level (alpha, α) necessary to reject H o. This is typically P < Rank your Menhinick s Indices from smallest to largest in a table, noting which index came from which habitat. Example: Table 1 shows 18 (imaginary) values for Menhinick s Indices from the two ponds mentioned before, silted (S) and clear (C). Table 2 shows the values ranked and labeled by pond type. Table 1. Menhinick s Indices Table 2. Ranked Menhinick s Indices for silted and clear ponds D silted D clear Rank Ranked D values Habitat S S S S S S C S S C C C S C C C C C Notice in the ranked table that if two values are the same, then the rank each one receives is the average of the two ranks. For example, value nine appears twice, at rank 6 and 7. Add the two ranks and divide by two to get their mean: 13/2 = 6.5. Each value is assigned their same, mean rank whenever there is a tie. Biodiversity Data Analysis 5

6 4. Assign points to each ranked value. Each silted rank gets one point for every clear rank that appears below it. Every clear value gets one point for every silted value that appears below it. For example, the first rank, 2(s) has 9 clear values below it, so it gets 9 points. Value 9(c) has 3 silted values below it, so it gets 3 points. Table 3. Points assigned to ranked D values in silted and clear ponds. Rank Ranked D Habitat Points values 1 2 S S S S S S C S S C C C S C C C C C 0 5. Calculate a U statistic for each category by adding the points for each habitat. U silted = = 75 U clear = = 6 Your final U value is the smaller of these two values. In this example our U value is 6. In general, the lower the U value, the greater the difference between the two groups being tested. (For example, if none of the D values overlapped, the U value would be zero. That means there is a large difference between the two groups: they do not overlap at all.) 6. You are now ready to move to the final step, determining whether to reject or fail to reject your null hypothesis. (Proceed to Section IV.) A video explanation of the Mann-Whitney U test procedure can be viewed here: B. Non-parametric test for multiple samples: Kruskal Wallis test We told you not to. But some teams just have to go that extra mile. Biodiversity Data Analysis 6

7 If your team is comparing more than two non-parametric data sets, a useful test, analogous to the ANOVA (Analysis Of Variance), is the Kruskal-Wallis test. This is well explained here: Kruskal Wallis: But you re on your own. We warned you. IV. Critical values for non-parametric statistics As you already know, a specific probability value linked to every possible value of any statistic, including the Mann-Whitney U statistic you just calculated. A. Critical values for the Mann-Whitney U statistic Remember that we have defined our significance level (α) as This implies that a correct null hypothesis will be rejected only 5% of the time, but correctly identified as false 95% of the time. A critical value of a statistic (e.g., your Mann-Whitney U statistic) is that value associated with a significance level of 0.05 or lower. The critical values for the Mann-Whitney U statistic are listed in Table 4. Compare your U value to those shown in the Table of Critical Values for the Mann-Whitney U (Table 4). Find the sample size (i.e., the number of Menhinick s Indices (D) you calculated) for each of your two habitats, and use the matrix to find the critical value for U at those two sample sizes. (For example, if you calculated 19 D values for one habitat and 17 for the other, then the critical value of the U statistic would be 99. This means that a U value of 99 or lower indicates rejection of the null hypothesis. In our example calculation, there were nine samples from two different habitats. In the Mann-Whitney U table, that corresponds to a critical value of 17. Our U statistic was 6, which is quite a bit lower than 17. This means that, if these were real data, we would reject the null hypothesis and fail to reject the alternative hypothesis. There is a significant difference in species richness between the clear and silted ponds. If your U value is lower than the critical value at the appropriate spot in the table, reject your null hypothesis. If your U value is greater than that in the table, fail to reject. B. Critical values for the Kruskall Wallis statistic If your team went crazy and decided to sample more than two different habitats, then your data analysis will be more complex. You will still use a non-parametric test, but it will be analogous to the parametric ANOVA, not the t-test. In this case, you will use the Kruskal Wallis test, as shown in the video linked above. Kruskal Wallis critical values are more complex, as they involve more than two data sets. Fortunately for us, J. Patrick Meyer (University of Virginia) and Michael A. Seaman (University of South Caroina) have made available a limited portion of a table of critical values they have calculated. These can be found here if your project involves either three or four data sets: The tables are not complete, but they do provide critical values for α levels of 0.1, 0.5, and You are unlikely to need other values; these will tell you whether to reject or fail to reject your null hypothesis. Biodiversity Data Analysis 7

8 Table 5. Critical values for the Mann-Whitney U statistic. Find the value that corresponds to the sample sizes of your two habitats. If your U value is smaller than that shown in the table, then there is less than 5% chance that the difference between your two habitats is due to chance alone. If your U value is smaller than the one shown in this table for your two sample sizes, reject your null hypothesis. If your U value is larger than that shown in the table, fail to reject your null hypothesis. (From The Open Door Web Site, Biodiversity Data Analysis 8

9 V. Project Completed. Is This the End? The study you are now completing is only the beginning of what could be a long-term research project to discover the various factors that affect biodiversity. The only thing you are determining now is whether or not there is a statistically significant difference between your two sample habitats. In other words, the research project you are now completing is a pilot Biodiversity Data Analysis 9

10 study. It establishes an observable fact (i.e., that there is or is not a difference in biodiversity between your two sample habitats). That fact should be subject to further investigation beyond what you have accomplished here. Although you may have established that there is or is not a difference in biodiversity between your two sample habitats, you still cannot definitively state why or why not there is a difference. To do that, you must move to the next step, which is to list as many competing hypotheses as possible as to why there is a difference (or even if your team has obtained negative results why there is not a difference, despite obvious differences in your two sample habitats). Each of these multiple hypotheses could form the basis for a research project that would take your team one step further towards discovering the reasons for your pilot study s observed result. You should be able to give a brief description of an experiment that could be designed to test each of your competing hypotheses. In your presentation, be sure to include a list of hypotheses that could explain your observed results. What factors differed between the two habitats that might cause differences in biodiversity? Would these factors affect the physiology of any organisms that lived there? Or would they simply be more hospitable to certain species and not others? When you consider your results, consider every aspect of your findings, and report anything you find intriguing enough to warrant further study. Science is not a one-project endeavor. Every finding of every research project can be seen as opening a new doorway to discovery of the most intimate mechanisms of life. Biodiversity Data Analysis 10

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